Partial derivative: d/dx x/sqrt(x^(2)+y^(2)+z^(2))?
d/dx x / sqrt(x^(2)+y^(2)+z^(2))
= [x^(2)+y^(2)+z^(2)-x^(2)] / [x^(2)+y^(2)+z^(2)]^(3/2)
But how? I can't get it to fall out :(
I got an answer earlier:
∂/ ∂x [ x/√(x^2 + y^2 + z^2 ]
= [√(x^2 + y^2 + z^2) - x (1/2)2x/√(x^2 + y^2 + z^2 ) ] /(x^2 + y^2 + z^2)
= [ x^2 + y^2 + z^2 - x^2 ] / (x^2 + y^2 + z^2)√(x^2 + y^2 + z^2)
= ( y^2 + z^2 ) / (x^2 + y^2 + z^2)^(3/2)
But I can't for the life of me see where the square root went in the first term of the derivative. I see how everything else is simplified, but how does that term suddenly become squared??
d/dx x / sqrt(x^(2)+y^(2)+z^(2))
= [x^(2)+y^(2)+z^(2)-x^(2)] / [x^(2)+y^(2)+z^(2)]^(3/2)
But how? I can't get it to fall out :(
I got an answer earlier:
∂/ ∂x [ x/√(x^2 + y^2 + z^2 ]
= [√(x^2 + y^2 + z^2) - x (1/2)2x/√(x^2 + y^2 + z^2 ) ] /(x^2 + y^2 + z^2)
= [ x^2 + y^2 + z^2 - x^2 ] / (x^2 + y^2 + z^2)√(x^2 + y^2 + z^2)
= ( y^2 + z^2 ) / (x^2 + y^2 + z^2)^(3/2)
But I can't for the life of me see where the square root went in the first term of the derivative. I see how everything else is simplified, but how does that term suddenly become squared??
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It all looks good:
∂/ ∂x [ x/√(x^2 + y^2 + z^2 ]
= [√(x^2 + y^2 + z^2) - x (1/2)2x/√(x^2 + y^2 + z^2 ) ] /(x^2 + y^2 + z^2)
then you multiply top and bottom of the fraction by √(x^2 + y^2 + z^2) to get
[ x^2 + y^2 + z^2 - x^2 ] / (x^2 + y^2 + z^2)√(x^2 + y^2 + z^2)
(there are two terms on the top that are multiplied, not just one)
∂/ ∂x [ x/√(x^2 + y^2 + z^2 ]
= [√(x^2 + y^2 + z^2) - x (1/2)2x/√(x^2 + y^2 + z^2 ) ] /(x^2 + y^2 + z^2)
then you multiply top and bottom of the fraction by √(x^2 + y^2 + z^2) to get
[ x^2 + y^2 + z^2 - x^2 ] / (x^2 + y^2 + z^2)√(x^2 + y^2 + z^2)
(there are two terms on the top that are multiplied, not just one)